Questions: Solve the inequality and write your answer in interval notation. -6+12 x<3(19 x+9)

Solution Steps

To solve the inequality \(-6 + 12x < 3(19x + 9)\), we need to follow these steps:

1. Distribute the 3 on the right-hand side.
2. Combine like terms.
3. Isolate the variable \(x\) on one side of the inequality.
4. Solve for \(x\).
5. Write the solution in interval notation.

First, we need to distribute the 3 to both terms inside the parentheses on the right-hand side of the inequality: \[ -6 + 12x < 3(19x + 9) \] \[ -6 + 12x < 3 \cdot 19x + 3 \cdot 9 \] \[ -6 + 12x < 57x + 27 \]

Next, we will move all terms involving \(x\) to one side of the inequality and constant terms to the other side. Subtract \(12x\) from both sides: \[ -6 + 12x - 12x < 57x + 27 - 12x \] \[ -6 < 45x + 27 \]

Now, we need to isolate \(x\) by moving the constant term on the right-hand side to the left-hand side. Subtract 27 from both sides: \[ -6 - 27 < 45x \] \[ -33 < 45x \]

To solve for \(x\), divide both sides by 45: \[ \frac{-33}{45} < x \] Simplify the fraction: \[ \frac{-33}{45} = \frac{-11}{15} \] \[ -\frac{11}{15} < x \]

The inequality \(-\frac{11}{15} < x\) can be written in interval notation as: \[ x \in \left( -\frac{11}{15}, \infty \right) \]

Final Answer